Very interesting bet. But nowhere do I read that the coins have
actually one head and one tail. What if they are tossing a two-headed
or two-tailed or a faceless coin?
Will the odds and probabilities change then? A lot of assumptions I see
:)
Best Regards,
Dario
>
> At 04:45 PM 10/18/98 -0600, Burgy wrote:
>
> >Brian wrote:
> >
> >"1. HHHHTTT
> >
> >2. THHTHTT
> >
> >3. TTTTTTT
> >
> >On which would you bet?"
> >
> >True -- that all are equally probable. But there is one more piece of
> >information
> >you gave us -- you said YOU HAD ACTUALLY TOSSED the coins.
> >
> >So since #2 is more typical, and since it is more likely you tossed
> >something
> >typical than not -- I'd bet on #2.
> >
> >What am I missing?
> >
>
> Hello Burgy. No, you are not missing anything.
>
> For those who may have missed it or don't remember,
> the quote above is from a wager that I found in a book
> by Piattelli-Palmarini. The quote was given because it
> closely relates to probability arguments one often sees
> in evolution/creation discussions. To refresh our memories,
> here's the bet:
>
> ===============================================
> I have just tossed a coin 7 times, and I ask you, who
> have not seen the result, to guess which of the three
> sequences below represents the sequence of my results.
> I guarantee that one of the sequences is genuine. If
> you don't get it right, you lose 10 dollars; if you
> win, you get 30. H stands for heads, and T for Tails.
>
> 1. HHHHTTT
>
> 2. THHTHTT
>
> 3. TTTTTTT
>
> On which would you bet? Let's think for a moment before
> going on.
>
> Experiments with a great many subjects have shown that
> the bets will be placed in the following order:2,1,3.
> The preference for the second sequence is very strong.
> But probability theory tells us that in seven tosses of
> a coin the probabilities are totally even, and we rationally
> should be quite indifferent to which of the three sequences
> we choose. The person who chooses 2 is prey to one of the
> most common cognitive illusions; she mistakes the most
> <typical> for the most <probable>.
> -- Piattelli-Palmarini, <Inevitable Illusions: How Mistakes of
> Reason Rule Our Minds>, John Wiley & Sons, 1994, p. 49-50.
> ====================================================
>
> So, both Burgy and myself seem to have fallen prey to a "common
> cognitive illusion", :) mistaking most typical for most probable.
>
> What does the author have in mind here? OK, clearly the
> probability of any specific sequence is the same, so if one
> had to guess what a result would be in advance, perhaps he
> has a point. But that isn't the way the bet works. So, why
> isn't the most typical the most probable? Perhaps the problem
> is (1) that one is stuck on the limitations of probability theory,
> having to deal only with specific outcomes and (2) the idea of
> "typical" is not very precisely defined. i.e., suppose we had to
> defend our conclusion to a skeptic. They say, "OK smart guy,
> prove that a typical result is the most probable outcome
> from a single trial". If all one has is probability theory then
> one seems to be stuck with answers like "isn't it obvious?",
> or, as Rafiki said to Simba "Look haaarder" ;-).
>
> In a recent post I mentioned results that have been developed
> only in the last thirty years or so (algorithmic information
> theory, AIT) which should lay this question to rest. In AIT
> one *is* able to precisely define what "typical" means and
> show that *any* result from a coin tossing experiment that
> contains *any* pattern is very highly unlikely.
>
> But suppose one doesn't know this result. What's the proper
> way to settle the dispute? Is it through unwavering faith in
> probability theory with condescending remarks about
> cognitive illusions? No, one should do an experiment. Repeat
> the wager many times and see who emerges with the most
> money. Now, its very obvious that Piattelli-Palmarini didn't
> actually perform this experiment. First of all, based upon the
> results from AIT he would clearly lose in the long run. Secondly,
> he really botched the odds. I could just guess randomly and
> still win money in the long run.
>
> I find this to be enormously interesting and entertaining
> considering the title of the book <Inevitable Illusions: How
> Mistakes of Reason Rule Our Minds>.
>
> Brian Harper
> Associate Professor
> Applied Mechanics
> The Ohio State University
>
> "It appears to me that this author is asking
> much less than what you are refusing to answer"
> -- Galileo (as Simplicio in _The Dialogue_)
--